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Properties

Label 4-312e2-1.1-c3e2-0-5
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 338.876338.876
Root an. cond. 4.290524.29052
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 4·5-s − 20·7-s + 27·9-s − 60·11-s − 26·13-s − 24·15-s − 84·17-s − 60·19-s − 120·21-s − 72·23-s − 210·25-s + 108·27-s − 124·29-s − 108·31-s − 360·33-s + 80·35-s + 36·37-s − 156·39-s − 52·41-s − 32·43-s − 108·45-s − 428·47-s − 134·49-s − 504·51-s + 380·53-s + 240·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.357·5-s − 1.07·7-s + 9-s − 1.64·11-s − 0.554·13-s − 0.413·15-s − 1.19·17-s − 0.724·19-s − 1.24·21-s − 0.652·23-s − 1.67·25-s + 0.769·27-s − 0.794·29-s − 0.625·31-s − 1.89·33-s + 0.386·35-s + 0.159·37-s − 0.640·39-s − 0.198·41-s − 0.113·43-s − 0.357·45-s − 1.32·47-s − 0.390·49-s − 1.38·51-s + 0.984·53-s + 0.588·55-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 338.876338.876
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 97344, ( :3/2,3/2), 1)(4,\ 97344,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1+4T+226T2+4p3T3+p6T4 1 + 4 T + 226 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+20T+534T2+20p3T3+p6T4 1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+60T+3114T2+60p3T3+p6T4 1 + 60 T + 3114 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+84T+10582T2+84p3T3+p6T4 1 + 84 T + 10582 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+60T+6526T2+60p3T3+p6T4 1 + 60 T + 6526 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+72T+14430T2+72p3T3+p6T4 1 + 72 T + 14430 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+124T−1586T2+124p3T3+p6T4 1 + 124 T - 1586 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+108T−16154T2+108p3T3+p6T4 1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−36T+42382T2−36p3T3+p6T4 1 - 36 T + 42382 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+52T+76666T2+52p3T3+p6T4 1 + 52 T + 76666 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+32T+150198T2+32p3T3+p6T4 1 + 32 T + 150198 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+428T+116242T2+428p3T3+p6T4 1 + 428 T + 116242 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−380T+258142T2−380p3T3+p6T4 1 - 380 T + 258142 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+1420T+882490T2+1420p3T3+p6T4 1 + 1420 T + 882490 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−1012T+708206T2−1012p3T3+p6T4 1 - 1012 T + 708206 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+844T+778238T2+844p3T3+p6T4 1 + 844 T + 778238 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+868T+888050T2+868p3T3+p6T4 1 + 868 T + 888050 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+60T+114886T2+60p3T3+p6T4 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−272T+993374T2−272p3T3+p6T4 1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1252T+1118698T2+1252p3T3+p6T4 1 + 1252 T + 1118698 T^{2} + 1252 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−572T+853306T2−572p3T3+p6T4 1 - 572 T + 853306 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−708T+1762390T2−708p3T3+p6T4 1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.95022448435788984664116694516, −10.32562671646902159028816148497, −9.993498419882439564928277611599, −9.734403784483738748995039249710, −8.911621503580280151647705856309, −8.842982570830159360008184520910, −8.028751454148506341239027958314, −7.72187164675514945245562756759, −7.34869108573832431349859057970, −6.73373998475477755736772247052, −6.05350067582724653057719832987, −5.64994998988630759394864214797, −4.55562015057282169677214473534, −4.45692626904635966156228758992, −3.33770158113213474685252645889, −3.26800874539872326447758293689, −2.13155032319669596782182198073, −2.06605078619900576989463972130, 0, 0, 2.06605078619900576989463972130, 2.13155032319669596782182198073, 3.26800874539872326447758293689, 3.33770158113213474685252645889, 4.45692626904635966156228758992, 4.55562015057282169677214473534, 5.64994998988630759394864214797, 6.05350067582724653057719832987, 6.73373998475477755736772247052, 7.34869108573832431349859057970, 7.72187164675514945245562756759, 8.028751454148506341239027958314, 8.842982570830159360008184520910, 8.911621503580280151647705856309, 9.734403784483738748995039249710, 9.993498419882439564928277611599, 10.32562671646902159028816148497, 10.95022448435788984664116694516

Graph of the ZZ-function along the critical line